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  "body": "\n## Statement\n\nLet $V$ be an inner product space with the [[induced norm|0MFJJV97RJ]] $\\norm{\\cdot}$, and let $x, y \\in V$.\nIf $V$ is a real space, then\n$$\\langle x, y \\rangle = \\frac{1}{4} \\left( \\norm{x + y}^2 - \\norm{x - y}^2 \\right).$$\nIf $V$ is a complex space, with the inner product linear in the first argument, then\n$$\\langle x, y \\rangle = \\frac{1}{4} \\sum_{k=0}^{3} i^k \\norm{x + i^k y}^2.$$\n\n## Remarks\n\nSo the norm determines the inner product.\nIn particular, a linear map that keeps norms also keeps inner products.\nWith the convention of linearity in the second argument, the factor $i^k$ moves to the other vector, and the complex formula changes accordingly.\n\n## Explanations\n\n### L1 human\n\nIf we know the lengths of all vectors, then we also know all inner products.\nIn the real case, we only need the lengths of the two diagonals $x + y$ and $x - y$.\nSo lengths determine angles.\n",
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  "name": "Polarisation identity",
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